Author:
Colarte-Gómez Liena,Costa Laura,Marchesi Simone,Miró-Roig Rosa M.,Salat-Moltó Marti
Abstract
AbstractIn this paper, we introduce the notion of a complete hypertetrahedral arrangement $${\mathcal {A}}$$
A
in $${\mathbb {P}}^{n}$$
P
n
. We address two basic problems. First, we describe the local freeness of $${\mathcal {A}}$$
A
in terms of smaller complete hypertetrahedral arrangements and graph theory properties, specializing the Mustaţă–Schenck criterion. As an application, we obtain that general complete hypertetrahedral arrangements are not locally free. In the second part of this paper, we bound the initial degree of the first syzygy module of the Jacobian ideal of $${\mathcal {A}}$$
A
.
Publisher
Springer Science and Business Media LLC
Reference14 articles.
1. Abe, T., Dimca, A., Sticlaru, G.: Addition-deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements. J. Algebraic Comb. 54(3), 739–766 (2020)
2. Dimca, A.: Curve arrangements, pencils, and Jacobian syzygies. Mich. Math. J. 66(2), 347–365 (2017)
3. Dolgachev, I.: Logarithmic sheaves attached to arrangements of hyperplanes. J. Math. Kyoto Univ. 47(1), 35–64 (2007)
4. Dolgachev, I., Kapranov, M.: Arrangements of hyperplanes and vector bundles on $${\mathbb{P}}^{n}$$. Duke Math. J. 71, 633–664 (1993)
5. Edelman, P.H., Reiner, V.: Free hyperplane arrangements between $$A_{n-1}$$ and $$B_{n}$$. Math. Z. 215, 347–365 (1994)